Harmonic Functions

A)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b},
L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane.

b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the exception of the linesegment from ia to ib.

c) Define h(z)=F(log(M(z))) for R(z)>0. Show that h is harmonic and that h(z) is greater than 0 and less than pi

d)Show that log(z-ic) is defined for R(z)>0 and any real number c. Prove that
|F(log(z-ic))|is less than pi/2 in this region

e) Prove that h(z)=F(log(z-ia) - log(z-ib))

f) Use the fundamental theorem of calculus to show that integral(from a to b) of
dt/z-it = i(log(z-ib)-log(z-ia))

g) Combine part e and f to show that
h(x+iy)=integral(from a to b) of xdt/(x^2 +(y-t)^2)=arctan((y-a)/x)-arctan((y-b)/x)

h)Intrpret part g geometrically by showing that h(z) measures the interior angle of the triangle with vertices ia, ib and z at the vertex z. What are the limits of h(z) as
R(z)->0 for F(z) in (a,b) and for F(z) not in [a,b]?

Solution Summary

Harmonic functions are investigated. The solution is detailed and well presented.